Question

Math

Posted 3 months ago

```
Heba was asked to find this integral using $u$-substitution:
$\int(-12 x-1) \sqrt{-6 x^{2}-x+1} d x$
How should Heba define $u$ ?
Choose 1 answer:
(A) $u=-6 x^{2}-x+1$
(B) $u=\sqrt{-6 x^{2}-x+1}$
(c) $u=-12 x-1$
(D) $u=\sqrt{x}$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 2

Looking at the integrand $(-12x-1)\sqrt{-6x^2-x+1}$, we notice that the derivative of $-6x^2-x+1$ is $-12x-1$, which is present outside the square root

step 3

Therefore, we should choose $u$ to be the expression inside the square root, $u = -6x^2-x+1$, so that $du$ will be proportional to $-12x-1 \, dx$

step 4

This choice will simplify the integral, as the substitution will cancel out the $-12x-1$ term, leaving us with an integral in terms of $u$

A

Key Concept

$u$-substitution

Explanation

In $u$-substitution, we look for a function within the integral whose derivative is also present. This allows us to simplify the integral by substituting $du$ for the corresponding terms in $dx$.

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